To study the long time asymptotic behaviors, usually we only use the \(x\) -part of Lax pair to analyze the initial value problem, and the \(t\) -part is used
to determine the time evolution of the scattering data for the integrable equations by inverse scattering transform method.
But unlike those of NLS and derivative NLS euqations [1]}, [2]}, [3]}, [4]}, the Lax pair (REF ) for SP equation has
singularities at \(z=0\)  and \(z=\infty \) . In order to control the behavior of solutions of (REF ) and construct the solution \(u(x,t)\)  of the SP equation (REF ), we need
use the \(t\) -part and the expansion of the eigenfunction as spectral parameter \(z\rightarrow 0\) . So we use two different transformations respectively to analyze these two singularities \(z=0\)  and \(z=\infty \) . Because in the case \(z=0\)  it has well property, we first consider this case.
